Combining Conceptual and Procedural Understanding in Math

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Singapore Math22

The fact of the matter is that everything in math is done for a reason, not merely because of a memorized rule. Richard Skemp acknowledged two approaches to math: instrumental and relational. Instrumental deals with the rote memorization of rules, algorithms, and procedures to arrive at a correct answer. Relational emphasizes connections between the algorithms, procedures, and concepts previously used while increasing their complexity to arrive at a correct answer.

A procedure can often be easier for teachers to give to students, and students can easily apply it in the moment. A teacher provides the rule to get the correct answer in a given situation—a particular word problem or an upcoming test—and the student uses it. This doesn’t incorporate the deeper understanding that the conceptual or relational does. With conceptual understanding, students can apply and adapt prior knowledge to new tasks, thereby making math applicable to more than just a particular word problem or upcoming test. With this approach, teachers can help students become confident, creative, and persistent learners.

Combining the Two Approaches

The fusion of conceptual with procedural understanding is not to the exclusion of mastery. In order to get to mastery, teachers can consider a concrete to pictorial to abstract approach based upon the work of Jerome Bruner. This enables students to develop conceptual understanding of concepts as well as procedures, which they then practice.

The practice should be varied and can include activities and games as well as independent work. It is through this practice that students become proficient, fluent, and flexible as the structure of math is discovered, rather than mere rules memorized and applied. By establishing this structure—and with the guidance of the teacher—students believe they are the ones developing the rules, thus making it their math and they are the master mathematicians.

Students should apply concepts and skills in various engaging situations as a vehicle to develop thinking skills and as the focus of learning math. Visual models from place value charts and fractions on a number line to ten frames and number bonds are effective visual tools employed as part of the concrete to pictorial to abstract approach.

The Singapore Math® Pedagogy

Bar modelling—the hallmark of the Singapore Math® approach—is a method that helps students visualize a situation and organize the known and unknown variables from a problem into a more accessible model. The bar model is a problem-solving tool that can be used from elementary through middle school and beyond. It even becomes a useful bridge to algebra.

How math is taught with the Singapore approach plays an important role in its proven success (TIMSS, PISA). Singapore Math® incorporates conceptual with procedural understanding, includes the concrete to pictorial to abstract approach, and emphasizes problem solving.

The views expressed in this article are those of the authors and do not necessarily represent those of HMH.


Interested in learning more about Singapore Math®, which combines conceptual and procedural understanding of concepts in the math classroom? Register here for our webinar, “Effective Math Instruction—What Have We Learned and Where Do We Go from Here?” presented by Laura Gifford and Christopher Coyne on Feb. 13 at 4-5 p.m. ET.

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